Chapter 9: Vector Calculus
Section 9.8: Divergence Theorem
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Example 9.8.6
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Apply the Divergence theorem to the vector field and , the region bounded by the paraboloid and the plane .
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Solution
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Mathematical Solution
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The divergence of F:
Implement the integral of over the interior of :
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To compute the flux through , note that there are two boundaries, the paraboloid, and the unit disk in the plane . To compute the flux through the paraboloid, note that on the paraboloid
If this be integrated over the unit disk, the result is
On the upper boundary (disk), the outward normal is , so , which becomes in the plane . Implementing the flux integral in polar coordinates gives
The total flux is then , the same value obtained for the volume integral of the divergence, as predicted by the Divergence theorem.
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Maple Solution - Interactive
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The Student VectorCalculus package is needed for calculating the divergence, but it then conflicts with any multidimensional integral set from the Calculus palette. Hence, the Student MultivariateCalculus package is installed to gain Context Panel access to the MultiInt command.
Initialize
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Tools≻Load Package: Student Vector Calculus
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Loading Student:-VectorCalculus
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Tools≻Tasks≻Browse: Calculus - Vector≻
Vector Algebra and Settings≻
Display Format for Vectors
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Press the Access Settings button and select
"Display as Column Vector"
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Display Format for Vectors
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Define the vector field F
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Enter the components of F in a free vector.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Conversions≻To Vector Field
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Context Panel: Assign to a Name≻F
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Obtain , the divergence of F, and represent it parametrically
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Common Symbols palette: Del and dot-product operators
Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻
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Obtain the volume integral of the divergence of F
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Tools≻Load Package:
Student Multivariate Calculus
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Loading Student:-MultivariateCalculus
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Write the name given to the divergence.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Multivariate Calculus≻Integrate≻Iterated
Complete the dialogs as per the figures below.
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Context Panel: Evaluate Integral
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There are two parts to the boundary of , the paraboloid and a unit disk in the plane . For the flux through the paraboloid, use a task template.
Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻3-D≻Through a Surface Defined over a Disk
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Flux through a Surface Defined over a Disk
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For the Vector Field:
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For the "open" surface , Maple uses the normal , where is a position-vector description of the paraboloid. This gives a normal that is upward, and hence inward for the closed region . Because is a closed region, the normal should be outward, and hence downward; the flux through the paraboloid is actually .
On the upper boundary (disk), the outward normal is , so , which becomes in the plane . The flux through this disk is given by the following calculation in which the integration is implemented in polar coordinates.
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Write the integrand.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Multivariate Calculus≻Integrate≻Iterated
Complete the dialogs as per the figures below.
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Context Panel: Evaluate Integral
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The total flux is then , the same value obtained for the volume integral of the divergence, as predicted by the Divergence theorem.
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Maple Solution - Coded
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Initialize
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Install the Student VectorCalculus package.
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Obtain , the divergence of F
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Use the int command to integrate the divergence of F over
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Use the Flux command to obtain the flux of F through the paraboloid
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For the "open" surface , Maple uses the normal , where is a position-vector description of the paraboloid. This gives a normal that is upward, and hence inward for the closed region . Because is a closed region, the normal should be outward, and hence downward; the flux through the paraboloid is actually .
On the upper boundary (disk), the outward normal is , so , which becomes in the plane . The integration of over the unit disk is simple enough to implement directly; the Flux command is a viable alternative.
The total flux is then , the same value obtained for the volume integral of the divergence, as predicted by the Divergence theorem.
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